Introduction to the Scales and Structures of Physics

Mass, length and time with nothing but our senses

So, we start with the most basic concepts of all in physics — mass, length and time. And I would like to start by asking you to imagine that we have no tools, no mathematics, no physics, no equations, but just our senses, nothing more than that. And we ask: what kind of masses, lengths and times can be perceived with just our senses and nothing more than that? So no instruments of any kind, and no analysis of any kind; just our bare senses.

Let us start with mass, for example, and ask — in terms of masses, we will use standard international units throughout — what is the smallest mass that you think you can estimate? By holding it, weighing it in this fashion, like this… what do you think is the smallest mass?

StudentA gram.

One gram. A gram, certainly, a gram you can tell. You can tell the difference between a gram and a kilogram, with our intuition. How about a nanogram and a picogram? Could you tell the difference? You could not do this; certainly, a fraction of a gram would be a safe estimate. So let us put this down — mass in kilograms — and say that you can start estimating something which is of the order of a gram or maybe a fraction of a gram, so that is 10⁻⁴ kilograms.

And what is the heaviest mass you think you could estimate? Ten kilograms? You could certainly tell the difference between 10 kilograms and 100 kilograms — certainly. How about 1000 kilograms?

StudentYou will not be able to lift it.

Well, you will not be able to lift it — that is a good point. And in fact, if I gave you a lump of metal and did not tell you its density, then you have no way of knowing whether it is 10,000 kilograms or 100,000 kilograms or 1,000,000 kilograms, at all. So whatever you can push, just about, is the upper limit; maybe 100 kilograms. Let us play safe and have another order of magnitude — 1000 kilograms. So that is 10³ kilograms, the upper limit of what you can do with just your senses.

What about length? Let us measure this in metres. What do you think is the smallest length you could perceive with a naked eye? A fraction of a millimetre; maybe half a millimetre or something like that. So we will play it safe once again: a millimetre is 10⁻³, and I say that with a very sharp resolving power, 10⁻⁴ metres.

And what is the longest distance that you could estimate, without instruments? Well, you could not tell the difference between the distance to a planet and the distance to a star with a naked eye, unless you have some other piece of information. And on a clear day, if you stand in a very clear place and look out from a mountain peak, you could perhaps see 10 kilometres. But you see, the point is, if you did not have other objects for reference, you have no way of estimating how far things are. If you had a blank wall with no texture on it, no signs, no way of distinguishing what the relative scales are and so on, then you have no way of knowing how far things really are. But perhaps 10 kilometres is a good estimate; you certainly cannot tell the difference between 1000 kilometres and 100 kilometres.

StudentSir, you are not allowed to run?

No, I am saying with your bare senses, and that is it. Of course, if you start running, and you say what endurance is and so on, that is a different story; but let us just say you are standing and you are trying to do an experiment to see how far you can see. Ten kilometres, safe thing; that is about 10⁴. These are just orders of magnitude — 10⁴ metres.

Why we do not notice our own blinking

And what about time? What is the smallest time that you could perceive, in seconds? Oh — the blink of an eyelid. Certainly a pulse is of the order of a second, but you can measure time scales smaller than that. The blink of an eyelid, that is about how much? A fraction of a second, 10⁻¹ seconds.

Incidentally, while we are talking about it, why do we blink? Certainly, you would like to clean the eye every now and then, you would like to have a layer. But how is it that when I blink — I am sorry, I am going to go off into digressions of this kind — how is it that when you blink, things do not go off?

We blink in a tenth of a second, and we blink maybe every 5 seconds or so. If every 5 seconds I switched off the lights in this room, it would be tremendously distracting to you. This can be done; this is a very easy experiment to do. You find it tremendously distracting if every 5 or 10 seconds the lights went off for a tenth of a second at a time — and that is what you do when you blink. But it does not happen when we look around. Why do you think that is the case?

StudentProbably the brain got used to it.

Close — it is a close answer. The processing section of the brain closes down simultaneously; otherwise you would not have this feature at all. Very cleverly designed; it would be tremendously distracting otherwise. So even the processing centre closes down. That is the reason why you do not feel that you are blinking, unless you do it consciously, of course: close your eye, and then of course you know that it is dark. But otherwise, a reflex blink you do not see as a shutting off of the lights at all, because even the processing is not done during this period.

Well, anyway, to come back — it is a tenth of a second here. What is the longest time that you can actually perceive without any instruments, without any external signal, without the seasons, without the stars, and so on? People have done this experiment. They have actually put people in cells with all the comforts of life provided; with uniform lighting so they cannot tell the difference between night and day; with the same bland food, so you cannot tell whether it is breakfast or lunch or dinner; provided in a cavity — you open up this recess and you get this food. Very familiar from hostel life, right? They are trying to get you used to this.

And then you are supposed to go and press a button every two hours or so, and pretty soon you realise that the experimental subject starts pressing the button every 3 hours or so, or every 4 hours or so. So she assumes that a certain amount of time — 2 hours — has elapsed, when really 3 or 4 or 5 hours have elapsed. And eventually even the rhythms of the body get knocked out, and gradually all the cycles change. I think this experiment has been done and people have gone to the stage where they think a day is like 50 hours long. So your perception of time goes off unless you have other indications, other longer cycles and so on; but even the physiological cycles change, everything changes along with this.

So I would say, maybe you can tell the difference between a month and a year, but you certainly cannot tell the difference beyond that. We will play it safe once again and say that the upper limit is of the order of 100 days, for instance. A day is about 10⁵ seconds, so 100 days is about 10⁷ seconds.

That is the world of middle dimensions, the world in which we have some intuition. But now we start asking: what does the universe do? On what scale is it operating? Then, of course, you are in for a tremendous surprise. This is the world of middle dimensions — I would like to put this in quotation marks — the macroscopic world in which we live, which we have got used to. There are other parameters, other physical variables like velocities and so on, which also have comparable numbers, but we have started with mass, length and time, and let us stick to that.

The scales the universe actually operates on

Now, what do you think is the smallest mass that we know of with our instruments, whatever we have detected? We have gone smaller than a proton. An electron is certainly less massive than a proton, so that is about 10⁻³⁰ kilograms. We do not know of particles with smaller masses than that at the moment. This is the mass of an electron, and there is a vast gap between these two.

On the upper limit, what is the largest mass you can think of? The mass of the known universe — certainly all the matter in the universe. We do not know how to estimate this; we do not know the size of the universe too well. Now, what do you think we should write here? How big do you think it is?

We could do the following: we could assume that we have a whole lot of stars and try to estimate the total mass of the stars. The total mass of all the stars would mean multiplying the number of stars by an average star — like the sun, for instance. Now, what is the mass of the sun? About 10³⁰ kilograms. We could estimate this, and remember, the way we would do this is always to assume that you are on a desert island with no information, no Wikipedia, no internet, and you have only sand to write on, and you need to make all these estimates for your survival. This is the way to learn any subject; you are compelled to do this. Then, of course, I could start by saying the sun is a hot ball of gas, I estimate the average density of a gas, multiply it by its size, and then so on and so forth. But 10³⁰ is about right.

And how many stars are there? About 10²². There are 10¹¹ galaxies, and about 10¹¹ stars in each galaxy; so that gives us 10⁵², and this is an estimate. Of course, you could do this another way: you could start by asking what the density of matter is — known matter, ordinary matter in the universe — and then multiply this by the radius of the known universe, by the size.

Now, what is the size of this universe? That is a very tricky question; it depends on what you mean by size. A very naive way of doing this would be to say, well, I know the age of the universe — and what is the age of the universe? 13.7 billion years. The captions read 18.7 here; 13.7 is what is meant, and is the figure used later in the lecture. The 0.7 is important; it is known, it is established, the first decimal point is known. You multiply that by a light year, because it is that old, and this would mean this is like the radius of the edge of the universe, stars receding from you at the speed of light, essentially. So this is a good estimate, and when you make that estimate it turns out that the answer is about 10⁵², once again.

There are other ways of doing this. Of course, this assumes that you are at the centre of the universe and the rest of it is simply expanding away from you, but that is not necessarily true at all. There are similar estimates which would tell you the comoving radius of the universe, and it is of the order of 40 billion light years or so — it is in the same ballpark. And therefore we will assume 10⁵² on this side. Of course, there is dark matter, there is dark energy and so on; forget about that. Whatever it is, for the purposes of this argument it is of the order of 10⁵², roughly.

The Planck length, mass and time

Now, what about length? What is the smallest length that you know of? Certainly experiments on electrons have told you that there is no structure whatsoever. We have done nuclear physics, we know the size of a nucleus is 10⁻¹⁵ metres — a femtometre. If you go beyond that, inside the nucleus etcetera. But we can conceive of an extremely small length from the three fundamental constants of nature — Planck's constant, the speed of light in vacuum, and Newton's gravitational constant. These are the three fundamental constants of nature and they are very, very important.

So you have Planck's constant, the speed of light, and Newton's gravitational constant. Those are the natural constants. Mass, length and time are something which we have created, in some sense, but the constants we have available in nature have different dimensionalities — not mass, length and time necessarily. What are the dimensions, the physical dimensions, of Planck's constant?

StudentJoules per second.

Those are units. Now what are the dimensions? It is energy multiplied by time; so it is ML²T⁻¹. And c is, of course, LT⁻¹, and G you can find from Newton's force equation. So with these three constants it is clear you can create three combinations which have dimensions of mass, length and time, and they would be called the Planck length, the Planck mass and the Planck time.

The Planck length turns out to be of the order of 10⁻³⁵ metres. Construct this as an exercise — a combination of h, c and G which has dimensions of length. So this is l_Planck, much, much smaller than the nuclear scale. And what is the largest length we can think of? The radius of the universe. We have various estimates for the size of the universe, but it is an upper bound; we do not even know if it is finite or not. Let us write this down as maybe 10²⁶ or 10³⁵ metres, very roughly — but it is enormous, as you can see.

What about time? Again, the Planck time is the smallest time we could construct, that we could think of as time itself, and that turns out to be of the order of 10⁻⁴² seconds. And what is the longest time that you could think of? The age of the universe, which is 13.7 billion years; therefore this is about 10¹⁰ years, which is 10¹⁰ multiplied by 10⁷, so 10¹⁷ seconds. But remember, these are orders of magnitude.

Now just think: this range here is invariably about 7 to 8 orders of magnitude, give or take 1 or 2 — a very tiny window. But nature is operating on a much, much bigger scale. It is operating on a mass scale which is like 80 orders of magnitude, to the extent we know about; a length scale which is at least 60 orders of magnitude, possibly much, much more, maybe even infinity; and a time scale which as of now is already about 60 orders of magnitude and could become much, much larger. And these are orders of magnitude, these are not just factors — you are not just doubling or tripling the range, you are actually multiplying by 10 each time. This is absolutely incredible, and it is just mind-blowing to see the range on which nature is operating.

Physical intuition is a myth

Therefore, I put it to you, any intuition which you develop to conduct your daily lives in this world of middle dimensions — there is no reason why that should continue to hold in this range; there is no reason at all. And the so-called understanding of classical physics, the intuition you develop and so on, the physical intuition is a myth; it is simply something which has been hardwired into your brains for reasons quite different from the understanding of nature. Therefore, there is no reason to expect that whatever law, whatever regularity you find in this range of orders of magnitude should continue to hold good within this range — and indeed, it does not.

So with the introduction of instruments — microscopes on the one hand, and telescopes on the other — you are really able to broaden the range in which you can probe nature, all the way from this side to that side. And simultaneously it turns out you need other tools; you need mathematical tools in order to understand how nature operates at this scale. So the real miracle is that you have a language, and you have the means to understand this range in some sense, even though it is not necessary for survival.

Why our senses stop where they do

Now, you could ask: why is it that I cannot perceive times smaller than this? Why is it that I cannot tell the difference between a picosecond and a nanosecond? Why do you think that is the case? You did not need it — you do not need it for survival, you do not need it evolutionarily for survival.

To put it very crudely — this is a delicate question, but you can give a very rough argument — after all it was all about survival, and what you needed was a reflex time that was sufficiently fast to ensure your survival. To put it in very graphic terms, certainly not to be taken literally: our primate ancestors had to make sure they did not fall down into the jaws of predators. The rate of fall is controlled entirely by gravity, and depending on that you needed time. When you let go of your mother ape and you started falling, you needed time for the muscles of the hand to send a message to your brain, and the brain to send a message back to the muscle to say — hold on, or you are going to fall down. A fraction of a second was sufficient for that; you did not need a picosecond, you did not need a femtosecond. So you did not have to waste brain power and neurons processing that information. You got hardwired into this world, and that was enough on this side.

All these things are guided by that. Mass, for example: you do not need to know the difference between a microgram and a picogram, but you do need to know the difference between a gram and a kilogram. And once again, in this metaphorical language, it was essential for our ancestors to know that you have to be much more effective throwing a rock at a predator rather than a leaf. So they had to know the difference between a gram and a kilogram; they did not need to know the difference between a microgram and a picogram. And that is why the brain did not waste any time trying to process information from this range or that range. And this is why we think we understand Newtonian physics — because you see masses, you see time, you can actually push these rocks around and so on.

But I put it to you that there is nothing intuitive about it. It took a long, long time to discover that when you push an object you change its velocity and not its position, and that you then change its position as a consequence of changing its velocity. After all, Newton's law says the force is proportional to the rate of change of momentum, or the velocity, not the position. Only bacteria, which are swimming in a Newtonian fluid at terminal velocity — only for them is the force proportional to the velocity, but not for us. So even Newton's law is counter-intuitive, very counter-intuitive, and its consequences can be equally dramatic.

So do not confuse facility in a certain range of mass, length and time with understanding of that range of mass, length and time — they are very different things. And it turns out that the language you need for understanding, for making predictions, is inherently mathematical. We do not know why; we do not know the deep reason why. We do not understand as yet why it is that our brains, which are hardwired evolutionarily for survival in a certain range of parameters, have been able to come out with a language and abstraction called mathematics which enables us to probe the rest of the region — and why it is that with the aid of these instruments which we have developed we are actually able to investigate a good bit of the range and understand it in some codified sense.

So that is a surprise. Not the fact that electrons behave like waves or like particles or whatever you might have heard; not the fact that Newton's inverse square law of gravitation, although universal, is actually an approximation — these are not surprises, really. It would be a surprise if it were not so. If everything was decided by a few simple equations, that would be a surprise. There is no reason why that should be so, and it is not so; it just is not so.

What this course will and will not cover

So classical physics is one portion of this range, which we will be able to uncover using rather simple rules — but not necessarily trivial rules; fairly complicated rules. And as the course goes along, I will show you that classical dynamics is actually quite intricate; it has a very, very precise structure, a very interesting structure. And quantum mechanics is a very non-trivial extension of this, and the relationship between the two is not yet fully understood. We do not fully understand what is going on in quantum mechanics, in a certain sense; but I will also point out that enough people have sensed that quantum mechanics is easier than classical dynamics, which is much more intricate mathematically.

And then, of course, there are other problems associated with the rest of the curriculum, such as statistical physics. What happens when you have large collections of objects? How do you understand them? Why do you need probabilistic concepts? Why do you at all need statistical concepts? This is something which is worth understanding, and it will turn out that we will acquire, at the end of this course, hopefully some perspective on why all this is happening, and where we stand today, and what kind of progress we could expect.

But for the moment we will stick to classical physics, in which we will switch off Planck's constant; and this curriculum also does not have much about gravitation, although I will mention it once in a while. So it will be non-relativistic, it will be non-quantum mechanical, and of course we will ignore gravitation for some part at least. So it turns out that you might as well eliminate G, set c equal to infinity, and h equal to 0 — and that would be the classical physics we are going to look at. But you must be aware that these are boundary conditions, these are limiting cases, and that the rest of it is really part of a much bigger whole.

Wherever necessary I will use orders of magnitude and estimates, and wherever we think we need to do something more rigorously we will work things out much more explicitly. I will try and do everything on the board, so that all equations are understandable. There may be cases where I might just quote a result — especially mathematical results — and say these are well known theorems and not bother about proving them; but we will understand them, we assume that rigorous proofs are available, and we will try to understand them in somewhat physical terms.

Physical does not mean mechanistic

Now, when I say physical, I would like to explain that I do not necessarily mean mechanistic. Everything need not be mechanistic at all. For example, electric and magnetic fields exist, we know that; they have classical limits, classical electric and magnetic fields exist. But you cannot give a mechanical model for them, not in terms of wheels, gears, pulleys and so on — this is not possible. They are fields; every point in space and time would have a field, and they may or may not be detectable or perceptible to you with one instrument or another, but that does not mean they do not have reality, that they do not exist because they are not hard objects like this. So the universe does not just consist of rigid bodies, does not just consist of classical waves; there is much more to it than that. Electromagnetic waves, for example, are already non-mechanical in that sense — they already go beyond your normal mechanical intuition, and yet they are completely classical. So you have to allow for larger possibilities as we go along. This is the sense in which I would say something is physical.

I assume also — might as well say this right at the beginning — that you are familiar with complex numbers. We will freely use as much mathematical machinery as we can. I have been asked this question in the past: if all measurements are real, why do we need complex numbers? That is a good question, but it should really be asked in class twelve, at that stage. It is just that you need these numbers; you need matrices, not real numbers necessarily; you need combinations of numbers, you need n-tuples, multiples of numbers and so on, and complex numbers are one such thing. So the word "imaginary number" does not mean anything as far as I am concerned. If you ask me what is the physical meaning of 2 + 3i, I would ask you what is the physical meaning of −3; for that matter, what is the physical meaning of three halves, or of 3? These are all abstractions, as you can understand, and we try to put them into correspondence with physical objects, and that is all that is being done. So it is simply a code, and this is the sense in which we would like to understand things.

The map of physics

Now, the first part of this course has to do with dynamics. But before I do dynamics, I would like to mention something along the lines of what I have already said, and what comes beyond dynamics — and this is the very famous picture, the very famous diagram, which I will try to reproduce here in some sense.

You have very, very slow velocities compared to the speed of light, and on scales much bigger than atomic length scales, microscopic length scales, you would have non-relativistic physics of macroscopic objects. So let us put a little box here and say this is roughly the region of non-relativistic classical mechanics, or Newtonian mechanics.

You could continue to remain non-relativistic but go into domains which are extremely small — atomic dimensions, for instance — and then you would have quantum mechanics. So here, joining it very crudely, you would have non-relativistic quantum mechanics.

On the other hand, you could look at fairly large objects but go to high velocities in this direction, and then you would have to take into account relativistic corrections. This would happen typically in astrophysics, for instance; certainly you have to take into account relativistic corrections there. So on this side you have relativistic mechanics, or relativistic physics. You keep going, you could include gravitational fields, very intense gravitational fields and so on. You would have special relativity, then you might have to make general relativistic corrections, and so on on this side, very schematically.

But then you could also go to very high speeds and very small objects, and then you would need relativistic quantum mechanics. And that is where a little bit of a surprise comes in.

Why relativistic quantum mechanics becomes field theory

It turns out that when you do relativistic quantum mechanics you could start with the kind of academic, very idealised problems we do in physics — just as in thermodynamics you start by studying the ideal gas. There is no ideal gas in nature, of course; everything interacts with everything else, but you use it as a useful model. Similarly, in quantum mechanics you might study, for example, the hydrogen atom — a single hydrogen atom in the universe. It is a very simple idealised model, and you could study its quantum mechanics as well: a particle orbiting around an attractive centre. You could do this quantum mechanically; you could do this relativistically.

But the moment you go to relativistic quantum mechanics, it turns out that a new phenomenon occurs, and this of course is the famous equivalence between matter and energy. It turns out that mass and energy are just two ways of saying more or less the same thing. Since matter and energy can be inter-converted, things like the number of particles are no longer a sacred concept when it comes to relativistic quantum mechanics.

Now, once that happens you can easily see that there is no such thing as relativistic quantum mechanics for a single particle; it is not likely to be consistent, simply because you could have many, many particles created — a single particle is very, very energetic — and then annihilated once again and reconverted to energy. This happens under certain constraints and conditions, but the principle is clear that it could happen. And therefore it turns out that relativistic quantum mechanics is intrinsically not really constructible, and what you need is the possibility that you can have any number of particles included in one single theory — and that leads you to relativistic quantum field theory. So really, this box is superseded by quantum field theory.

And that has turned out to be the most successful language of all: necessarily very intricate, very involved, but in principle it explains all that we know so far. It is like the culmination, the ultimate theory at this level of understanding that we have now of all the universe around us.

But there are very, very important gaps. One of them, of course, is familiar to you from reading popular literature: we have found no way in which you can consistently combine relativistic quantum field theory with gravitation. We do not yet know how to quantise gravity — that is one problem. The other problem is that the theory we have now for elementary particles at the fundamental level, called the Standard Model of particle physics, is itself an ad hoc theory. It is very obvious from the fact that it has many undetermined parameters, many parameters which you put in by hand, many constants. It is clear it cannot be a final theory of particle physics; there must be an underlying structure and we do not know what that is. So it is still incomplete. But the day is young — this whole thing is about a 100 years old, as you can see; it started about 400 years ago to reach this fairly fast. The day is young; there is a lot more to come.

Effective theories and the carburettor

But it is a good idea to have this perspective — that this is where it is at the moment, but you do not necessarily need all the complications of relativistic quantum field theory if you want to look at sub-portions of this. The whole thing is layered in such a way that, depending on the regime of physical parameters, depending on the physical problems we are looking at, you might find it sufficient to use an effective theory; you do not have to use first-principles theories all the time.

If you would like to design a better carburettor for your car, there is no reason why you should know the underlying organic chemistry of the fuel; there is even less reason to know that each molecule is made up of atoms, and that each atom in fact is made up of a nucleus and electrons, and that inside the nucleus there are nucleons — the neutrons and protons — and inside the neutrons there are quarks, and they are held together by things called gluons. It is not necessary for you to know that in order to design a better carburettor, although it is true.

So this business of reductionism should also be carefully examined. While you would like to have reductionism, to go to first causes in order to understand things from very basic principles, you must realise that at every level of organisation there is a set of effective laws, and this is really all you need.

The boundary between two regimes is very interesting. Always, these boundaries are interesting. Where classical mechanics stops and quantum mechanics starts — is it a fuzzy boundary, is it sharp? These are very interesting questions. Where non-relativistic physics stops and relativistic physics starts — again a very interesting question. On the other hand, inside a domain you could have an effective theory. This is what happens if you study elementary chemistry: if you would like to understand reactions, you have a law called the law of mass action. You can derive the law of mass action from more fundamental considerations, but you do not effectively need to do this if you want to understand chemical reactions in the large.

Emergent properties

And above all, you must recognise that when you take a whole lot of objects and put them together, the end product may be greater than the sum of the parts. There might be properties which come out for a collection which do not exist in each individual component. Individual atoms do not have colours, but when I put a sufficient number of atoms together into an object, it acquires a colour, for example. And this is a property that emerges due to the fact that you have a collection. And then a very interesting question is: at what stage does the colour emerge? And that is true for every one of these physical properties.

Individual water molecules do not do anything very interesting, but when you put a large number of them together it turns out that, depending on the external conditions, they could exist in different phases. The same interaction between two water molecules, under different conditions, could give you either ice or steam or water. So clearly this is a property of a collection and not a property of an individual molecule — that is another crucial thing we must bear in mind. There are these properties, called emergent properties, which emerge when you put a lot of things together, and one of our preoccupations is going to be with these properties. So this is another thing I would like to write down here — emergent properties, and effective models. I will use these phrases once in a while to mean precisely what I have just explained.

For example, Newtonian mechanics is an effective model in a certain regime of parameters — of length, mass, time, velocities, angular momenta and so on. So is quantum mechanics, as far as we know; it is possible that one day it is subsumed into a larger theory or a larger framework. But the difference between physical laws and mathematical laws is precisely this: whereas mathematical laws, once you lay the axioms down, would appear to be applicable in some absolute sense, physical laws are always applicable in some range of physical parameters.

Is the electron a wave or a particle?

You go beyond the range, and the law may or may not extrapolate in a smooth way, and this is also something which one should bear in mind, because this is very often lost sight of. That is what leads to questions of the kind: is the electron a wave or a particle? As we go along, I will convince you that the question itself is meaningless. An electron is what it is, and what properties it displays; in fact, that is going to be your attitude.

If I ask you what this piece of chalk is, every description, every definition that you think you give for what this piece of chalk is will be a statement of one of its properties. So I am not going to worry about what this chalk really is, and I will use the word — the piece of chalk — for a pre-agreed-upon collection of properties of this object. So that is going to be shorthand for a collection of properties which we have agreed upon. And then, of course, you see there is no difficulty with defining any object.

An electron is shorthand for the collection of its properties; it might turn out that these properties would depend on how you probe them, and it does in this bigger range that I talked about — and therefore I have no conflict at all. And then the question whether things like electrons are waves or particles becomes a question of semantics; it becomes a question of the failure of the words wave and particle to apply in that regime. In an unambiguous manner I need another language, but I have one: it is called quantum mechanics.

So I do not regard this wave–particle duality as a mystery. I just think it is a failure of ordinary language. And of course it would be a very unhappy situation if I did not have a language instead of ordinary language; but fortunately for us we have discovered that there is such a language, and we will use that language when the time comes.

So I hope you got our philosophy, right? The rest of it is not going to be so descriptive; it is going to get considerably more quantitative. Let me stop once again, and pause, and ask you — is there any comment or question that you would like to ask now?

Question: why do h, c and G give lower limits?

StudentSir, why is it that the combination of h, c and G gives the lower limits — the Planck length and Planck time?

It is a wonderful question, and a deep question: why is it that the combination of these three gives the lower limits of what we know, and not the upper limits and so on? This is not a very serious problem, in the sense that I do not know the upper limits; I do not know, for example, if the universe is finite or not, if it is unbounded. But what I do know is that there is a fundamental velocity — the speed of light; there is a fundamental quantum of action — Planck's constant; and there is G.

Now, in the case of Planck's constant, it turns out that this actually describes certain fluctuations, a certain indeterminacy, which you are familiar with in the guise of the uncertainty principle. And its numerical value is such that it happens to be applicable in the range of the very small; this is where fluctuations would play a role. Very rough answer, but we will get more precise answers as we go along.

Now, of course, it is not always true that it gives a lower bound. If I look at mass, for example — and I was careful not to use this, I did not use the Planck mass — the Planck mass, if I calculate it, turns out to be of the order of 10⁻⁵ grams. It is enormous compared to elementary particles; it is huge. So we believe Planck's constant has something to do with the ultimate reductionist level, with fluctuations at some microscopic, sub-microscopic level, and you would expect the mass also to do this — but the actual masses of the elementary particles that we know are often much, much smaller. So in a sense the Planck mass is a very large mass, a huge mass. Of course, if we put objects together you get much larger masses, like people, and galaxies, and collections of galaxies and so on, and it could be infinite on the other hand.

So it is not always true that it gives the lower limit. But in the case of length and time it does, because of a deeper reason: we believe that the Planck length and Planck time are in fact the length and time scales on which the concept of space-time as a continuum itself breaks down. We believe that below the Planck length and below the Planck time, thinking of time and length as continuous objects is itself suspect. We think the structure of space-time itself could be very different from what we know of it on much longer time scales and length scales; this is where quantum fluctuations in space and time themselves would start playing a role, and therefore the meaning of space, the meaning of length, the meaning of time is not very clear.

So that is why, on the lower end, simply because below that it is fluctuation-dominated, and above that in some sense these fluctuations are smoothed out, and on much larger scales you do not see these fluctuations at all. It is like saying that if I give you a piece of paper and tear it, the edge is jagged; but then, of course, you could look at it from sufficiently far away and it looks quite like a straight line, and as I go closer and closer I start seeing the fluctuations more and more. So it is at the lower level that you start seeing fluctuations. That is a very crude answer, but that is roughly what it is.

Question: how many molecules make a phase?

Any other thoughts? … This is precisely the point. I gave this example of water going into ice or steam or liquid water. It is quite clear that you need a sufficiently large number of molecules for this to happen, for you to be able to distinguish these phases. It is equally clear that the phase of water — whether it is solid, liquid or gas — is not a property of a single molecule; that is obvious. It is the same molecule; even the interaction between the molecules is exactly the same, and yet when you put an aggregate together it exists in these three phases.

The interesting question is: how many should I put together before I can tell whether it is liquid or gas and so on? Can I have an ice crystal which has only 10 water molecules, or 50, or 100? That is a much, much harder question. It turns out that there is no clear boundary in this sense; you could go on subdividing matter and then there comes a stage when you lose the concept of the phase. It happens fairly smoothly in most cases. We will talk a lot more about this when I discuss short-range order and long-range order, and liquids also.

A very rough answer would be: if you take, for example, water at low temperatures but before it freezes, you discover the system is trying to become crystalline. So neighbouring molecules are arranging themselves in a regular array, but then it is perturbed by thermal fluctuations and it dissipates. But as you lower the temperature the thing becomes more sluggish and eventually clicks into place as a crystal — but you do need a collection for this.

And the study of emergent properties is, of course, very, very vibrant. Phase transition is just one example of it, and there are many, many other such properties which are very interesting; it is also called collective behaviour, and it has many other terms — coherent structures and so on and so forth. Another example, for instance: if you took individual photons, nothing much happens; but if you put them all together in the right conditions they could get coherent and produce a laser beam — and that is not a property of a single photon. So many delicate things happen simply because you have these collections, and the collections could be extremely weakly interacting but still produce order. A very crude example would be: if there is a distraction at the other end of the room and just a few people in that corner start looking there, only their neighbours are influenced; they start looking there, and pretty soon that propagates. So even if you do only what your neighbours do, you can have very long-range coherence even though the interaction is short-range — that is an emergent property.